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Asymptotic properties of functionals of increments of a continuous semimartingale with stochastic sampling times

Michael Levine, Xiaoguang Wang, Jian Frank Zou

Abstract

This paper is concerned with asymptotic behavior of a variety of functionals of increments of continuous semimartingales. Sampling times are assumed to follow a rather general discretization scheme. If an underlying semimartingale is thought of as a financial asset price process, a general sampling scheme like the one employed in this paper is capable of reflecting what happens whenever the financial trading data are recorded in a tick-by-tick fashion. A law of large numbers and a central limit theorem are proved after an appropriate normalization. One application of our result is an extension of the realized kernel estimator of integrated volatility to the case of random sampling times.

Asymptotic properties of functionals of increments of a continuous semimartingale with stochastic sampling times

Abstract

This paper is concerned with asymptotic behavior of a variety of functionals of increments of continuous semimartingales. Sampling times are assumed to follow a rather general discretization scheme. If an underlying semimartingale is thought of as a financial asset price process, a general sampling scheme like the one employed in this paper is capable of reflecting what happens whenever the financial trading data are recorded in a tick-by-tick fashion. A law of large numbers and a central limit theorem are proved after an appropriate normalization. One application of our result is an extension of the realized kernel estimator of integrated volatility to the case of random sampling times.

Paper Structure

This paper contains 6 sections, 6 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. Model Set-up
  3. Laws of large numbers (LLNs) for increments of functions of semimartingales
  4. Main central limit theorem
  5. Discussion
  6. Acknowledgements

Key Result

Theorem 3.2

Let $f$ be a continuous function on $R^k$ for some $k \geq 1$, which satisfies for some $p > 0$ and $K_0$. Define Then,

Theorems & Definitions (8)

  • Definition 3.1
  • Theorem 3.2
  • Theorem 4.1
  • Remark 4.2
  • Lemma 4.3
  • Lemma 4.4
  • Lemma 4.5
  • Lemma 4.6